SAT Counting Principles: Permutations, Combinations, and the Fundamental Counting Method

The Fundamental Counting Principle states: if one event can happen in m ways and a second independent event can happen in n ways, both together can happen in m×n ways. This extends to any number of independent choices by multiplying all possibilities together. For example, 4 shirt choices and 3 pants choices gives 4×3=12 outfit combinations.

The key is independence: each choice must not reduce the options for the next. If you are choosing a 3-digit PIN from digits 0 to 9 with no repetition, the first digit has 10 choices, the second has 9, and the third has 8, giving 10×9×8=720 PINs. Any time the phrase "no repetition" appears in a counting problem, subtract one from the available choices at each subsequent step rather than keeping the count constant.

A permutation counts arrangements where the order of selection matters. The formula for choosing r items from n in a specific order is P(n,r)=n!/(n-r)!. Awarding first, second, and third place among 8 runners is P(8,3)=8×7×6=336. You use a permutation rather than a combination because first place is different from second place; the assignment is ordered.

Practice prompt: in how many ways can a president, vice-president, and secretary be chosen from 10 club members? Since roles are distinct, P(10,3)=10×9×8=720. Any time you are assigning distinct titles, rankings, or sequential positions, you are counting a permutation. If switching two people's assignments changes the outcome (A as president and B as secretary differs from B as president and A as secretary), you need a permutation, not a combination.

A combination counts selections where the order does not matter. The formula is C(n,r)=n!/(r!×(n-r)!). Choosing 3 students from 10 to form a committee (with no hierarchy of roles) gives C(10,3)=120. This is smaller than P(10,3)=720 because combinations divide out all duplicate orderings: selecting A, B, C is the same as selecting C, B, A for an unranked committee.

The distinction between permutation and combination is tested directly. Ask: does switching two selected items produce a different outcome? If selecting A, B, C gives the same result as C, B, A, use a combination. If they differ, use a permutation. The phrase "how many groups" or "how many committees" signals a combination, while "how many arrangements" or "how many ways to assign" signals a permutation.

Three common counting errors: (1) using a permutation when order does not matter, inflating the answer by a factor of r!; (2) forgetting to account for repeated elements, which reduces the count; (3) using the complement method incorrectly for "at least one" problems. For "at least one" problems, count the complement (none of the desired outcome) and subtract from the total number of arrangements or selections.

Use a 5-day drill: each day solve five counting problems and label each as "counting principle," "permutation," or "combination" before computing. Aim to label each problem in under 5 seconds before any arithmetic. Labeling the type before computing builds the decision habit that prevents applying the wrong formula under test-day time pressure.